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1.
Session – 7
Measures of Dispersions
The measures of Central Tendency alone will not exhibit various
characteristics of the frequency distribution having the same total frequency. Two
distribution can have the same mean but can differ significantly. We need to know
the extent of variation or deviation of the values in comparison with the central value
or average referred to as the measures of central tendency.
Measures of dispassion are the ‘average of second order’. The are based on the
average of deviations of the values obtained from central tendencies x , Me or z. The
variability is the basic feature of the values of variables. Such type of variation or
dispersion refers to the ‘lack of uniformity’.
Definition: A measure of dispersion may be defined as a statistics signifying the
extent of the scatteredness of items around a measure of central tendency.
Absolute and Relative Measures of Dispersion:
A measure of dispersion may be expressed in an absolute form, or in a relative
form. It is said to be in absolute form when it states the actual amount by which the
value of item on an average deviates from a measure of central tendency. Absolute
measures are expressed in concrete units i.e., units in terms of which the data have
been expressed e.g.: Rupees, Centimetres, Kilogram etc. and are used to describe
frequency distribution.
A relative measures of dispersion is a quotient by dividing the absolute
measures by a quality in respect to which absolute deviation has been computed. It is
as such a pure number and is usually expressed in a percentage form. Relative
measures are used for making comparisons between two or more distribution.
Thus, absolute measures are expressed in terms of original units and they are
not suitable for comparative studies. The relative measures are expressed in ratios or
percentage and they are suitable for comparative studies.
Measures of Dispersion Types
Following are the common measures of dispersions.
a. The Range
b. The Quartile Deviation (QD)
c. The Mean Deviation (MD)
d. The Standard Deviation (SD)
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2.
Range
‘Range’ represents the differences between the values of the extremes’. The
range of any such is the difference between the highest and the lowest values in the
series.
The values in between two extremes are not all taken into consideration. The
range is an simple indicator of the variability of a set of observations. It is denoted by
‘R’. In a frequency distribution, the range is taken to be the difference between the
lower limit of the class at the lower extreme of the distribution and the upper limit of
the distribution and the upper limit of the class at the upper extreme. Range can be
computed using following equation.
Range = Large value – Small value
L arg e value − Small value
Coefficient of Range =
L arg e value + Small value
Problems
1. Compute the range and also the co-efficient of range of the given series of state
which one is more dispersed and which is more uniform.
Series – I – 9, 10, 15, 19, 21 Series – II – 1, 15, 24, 28, 29
R = LV – SV = 21 – 9 = 12 R = LV – SV = 29 – 1 = 28
12 12 R 28
CR = = = 0.4 CR = = = 0.933
21 + 9 30 LV + SV 30
Series I is les dispersed and more uniform
Series II is more dispersed and less uniform
Evaluating Criteria
i. Less the CR is less dispersion
ii. More the CR is less uniform
Range Merits
i. It is very simplest to measure.
ii. It is defined rigidly
iii. It is very much useful in Statistical Quality Control (SBC).
iv. It is useful in studying variation in price of shars and stocks.
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3.
Limitations
i. It is not stable measure of dispersion affected by extreme values.
ii. It does not considers class intervals and is not suitable for C.I. problems.
iii. It considers only extreme values.
2. Find range of Co-efficient of range from following data.
A: 10 11 12 13 14
B: 40 41 42 43 44
C: 100 101 102 103 104
Series - I Series – II Series – III
R =LV – 3m
= 14 – 10 R = 44 - 40 R = 104 - 100
= 4 = 4 = 4
R R R
CR = CR = CR =
LV + SV LV + SV LV + SV
4 4 4
= = =
24 84 204
= 0.166 = 0.0476 = 0.0196
Series III is less dispersed and more uniform
Series I is more dispersed and less uniform
3. Compute range and coefficient of range for the following data.
x: 6 12 18 24 30 36 42
f: 20 130 16 14 20 15 40
Range = LV – SV = 42 – 6 = 36
R 36
CR = = = 0.75
LV + SV 48
3
4.
Quartile Deviation
Quartile divides the total frequency in to four equal parts. The lower quartile
Q1 refers to the values of variates corresponding to the cumulative frequency N/4.
Upper quartile Q3 refers the value of variants corresponding to cumulative
frequency ¾ N.
1
Quartile deviation is defined as QD = (Q3 – Q1). In this quartile Q2 as it
2
N
corresponds to the value of variate with cumulative frequency is equal to c.f. = .
2
1
a) QD = (Q3 – Q1)
2
Q 3 − Q1
b) Relative measure of dispersion coefficient of QD =
Q 3 + Q1
Problems
1. Find quartile deviation and coefficient of quartile deviation for the given grouped
data also compute middle quartile.
Class f
1 – 10 3
11 – 20 16
21 – 30 26
31 – 40 31
41 – 50 16
51 – 60 8
Σf = N = 100
Class f Cf
1 – 10 3 3
11 – 20 16 19
21 – 30 26 45 Q1 Class
31 – 40 31 76 Q2 & Q3 Class
41 – 50 16 92
51 – 60 8 100
N = 100
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8.
• It is very easy to compute
• It is not affected by extreme values of variable.
• It is not at all affected by open and class intervals.
Demerits of Quartile Deviation
• It ignores completely the portions below the lower quartile and above the upper of
quartile.
• It is not capable for further mathematical treatment.
• It is greatly affected by fluctuations in the sampling.
• It is only the positional average but not mathematical average.
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